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arXiv · 2106.07124

Self-orthogonal codes over a non-unital ring and combinatorial matrices

Minjia Shi·Shukai Wang·Jon-Lark Kim·Patrick Solé
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Abstract

There is a local ring $E$ of order $4,$ without identity for the multiplication, defined by generators and relations as $E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle.$ We study a special construction of self-orthogonal codes over $E,$ based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over $E,$ and Type IV codes, that is, quasi self-dual codes whose all codewords have even Hamming weight. All these codes can be represented as formally self-dual additive codes over $\F_4.$ The classical invariant theory bound for the weight enumerators of this class of codesimproves the known bound on the minimum distance of Type IV codes over $E.$

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Minjia Shi, Shukai Wang, Jon-Lark Kim, Patrick Solé. 2021-06-14. Self-orthogonal codes over a non-unital ring and combinatorial matrices. https://arxiv.org/abs/2106.07124

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